Even when there is no drift, so that μ = 0, the expected value of the lognormal distribution is affected by the volatility.
Thus, to simulate paths with truly zero drift, we have to subtract .5*σ² from the drift.
So now we have learned something many people overlook or forget: when you are simulating lognormal returns, you always have to subtract away half the variance from the drift. This is far and away the most common mistake in simulating returns.
Just take those 1,000 paths we simulated and compute the average
Also see: Lessons From The .50 Delta Option (Moontower)